# nLab cotangent bundle

Contents

### Context

#### Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

tangent cohesion

differential cohesion

$\array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& ʃ &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }$

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

# Contents

## Idea

Given a differentiable manifold $X$, the cotangent bundle $T^*(X)$ of $X$ is the dual vector bundle over $X$ dual to the tangent bundle $T x$ of $X$.

A cotangent vector or covector on $X$ is an element of $T^*(X)$. The cotangent space of $X$ at a point $a$ is the fiber $T^*_a(X)$ of $T^*(X)$ over $a$; it is a vector space. A covector field on $X$ is a section of $T^*(X)$. (More generally, a differential form on $X$ is a section of the exterior algebra of $T^*(X)$; a covector field is a differential 1-form.)

Given a covector $\omega$ at $a$ and a tangent vector $v$ at $a$, the pairing $\langle{\omega,v}\rangle$ is a scalar (a real number, usually). This (with some details about linearity and universality) is basically what it means for $T^*(X)$ to be the dual vector bundle to $T_*(X)$. More globally, given a covector field $\omega$ and a tangent vector field $v$, the paring $\langle{\omega,v}\rangle$ is a scalar function on $X$.

Given a point $a$ in $X$ and a differentiable (real-valued) partial function $f$ defined near $a$, the differential $\mathrm{d}_a f$ of $f$ at $a$ is a covector on $X$ at $a$; given a tangent vector $v$ at $a$, the pairing is given by

$\langle{\mathrm{d}_a f, v}\rangle = v[f] ,$

thinking of $v$ as a derivation on differentiable functions defined near $a$. (It is really the germ at $a$ of $f$ that matters here.) More globally, given a differentiable function $f$, the de Rham differential $\mathrm{d}f$ of $f$ is a covector field on $X$; given a vector field $v$, the pairing is given by

$\langle{\mathrm{d}f, v}\rangle = v[f] ,$

thinking of $v$ as a derivation on differentiable functions.

One can also define covectors at $a$ to be germs of differentiable functions at $a$, modulo the equivalence relation that $\mathrm{d}_a f = \mathrm{d}_a g$ if $f - g$ is constant on some neighbourhood of $a$. In general, a covector field won't be of the form $\mathrm{d}f$, but it will be a sum of terms of the form $h \mathrm{d}f$. More specifically, a covector field $\omega$ on a coordinate patch can be written

$\omega = \sum_i \omega_i\, \mathrm{d}x^i$

in local coordinates $(x^1,\ldots,x^n)$. This fact can also be used as the basis of a definition of the cotangent bundle.

## Properties

### Symplectic structure

Every cotangent bundle $T^\ast X$ carries itself a canonical differential 1-form

$\theta \in \Omega^1(T^* X)$

with the property that under the isomorphism

$j \;\colon\; \Gamma(T^* X) \stackrel{\simeq}{\to} \Omega^1(X)$

between differential 1-forms and smooth sections of the cotangent bundle we have for every smooth section $\sigma \in \Gamma(T^* X)$ the identification

$\sigma^* \theta = j(\sigma)$

between the pullback of $\theta$ along $\sigma$ and the 1-form corresponding to $\sigma$ under $j$.

This unique differential 1-form $\theta \in \Omega^1(T^* X)$ is called the Liouville-Poincaré 1-form or canonical form or tautological form on the cotangent bundle.

The de Rham differential $\omega \coloneqq d \theta$ is a symplectic form. Hence every cotangent bundle is canonically a symplectic manifold.

On a coordinate chart $\mathbb{R}^n$ of $X$ with canonical coordinate functions denoted $(x^i)$, the cotangent bundle over the chart is $T^\ast \mathbb{R}^n \simeq \mathbb{R}^n \times \mathbb{R}^n$ with canonical coordinates $((x^i), (p_j))$. In these coordinates the canonical 1-form is (using Einstein summation convention)

$\theta = p_i d x^i$

and hence the symplectic form is

$\omega = d p_i \wedge d q^i \,.$

Last revised on November 23, 2017 at 07:48:04. See the history of this page for a list of all contributions to it.